Porisma

Worked solution

Distinguish a sample count from the position of a final success

Rose and Violet contest each other for the post of President of the Student Council of Floral Junior College. A pre-election survey indicates that Rose has the support of 45% of the school population while Violet has the support of the remaining students.

(i) Find the probability that Rose has fewer than 12 supporters in a group of 30 randomly selected students. marks

(ii) If the students are asked to queue up to cast their votes one at a time, find the probability that the 16th student is the 10th student who votes for Violet. marks

[2010/DHS/BT2/12(a)]

CJC · Review Problems · Question 3

Answer

(i) 0.2330.233; (ii) 0.1050.105.

(i)

Using the source binomial model, let X∼B(30,0.45)X\sim B(30,0.45) count Rose supporters. Fewer than 12 means X≤11X\leq11, so P(X<12)=P(X≤11)≈0.2326871123=0.233P(X<12)=P(X\leq11)\approx0.2326871123=0.233 to 3 significant figures.

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(ii)

The first 15 voters must contain exactly nine Violet votes, and the 16th must also vote for Violet. Let Y∼B(15,0.55)Y\sim B(15,0.55). By the source independence model, the probability is P(Y=9)(0.55)=(159)(0.55)10(0.45)6≈0.1052703482=0.105P(Y=9)(0.55)=\binom{15}{9}(0.55)^{10}(0.45)^6\approx0.1052703482=0.105 to 3 significant figures.

How did this part go?

Common mistakes

Exactly ten Violet votes among sixteen does not force the sixteenth to be the tenth.

Solution notes

The source assumes approximately independent selections/votes with fixed support probabilities, and uses the survey proportions as voting probabilities.

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